Sr Examen
Integral de sin(pi*n*x/l)*t*(x^3)*l^2 dx
Solución
Ha introducido
[src]
1 / | | /pi*n*x\ 3 2 | sin|------|*t*x *l dx | \ l / | / 0
$$\int\limits_{0}^{1} l^{2} x^{3} t \sin{\left(\frac{x \pi n}{l} \right)}\, dx$$
Integral(((sin(((pi*n)*x)/l)*t)*x^3)*l^2, (x, 0, 1))
Respuesta (Indefinida)
[src]
/ // 0 for n = 0\ \
| || | |
| || // 3 /pi*n*x\ 2 /pi*n*x\ 2 /pi*n*x\ \ | |
| || || 2*l *sin|------| l*x *sin|------| 2*x*l *cos|------| | | |
/ | || || \ l / \ l / \ l / | | // 0 for n = 0\|
| | || ||- ---------------- + ---------------- + ------------------ for n != 0| | || ||
| /pi*n*x\ 3 2 2 | || || 3 3 pi*n 2 2 | | 3 || /pi*n*x\ ||
| sin|------|*t*x *l dx = C + t*l *|- 3*|<-l*|< pi *n pi *n | | + x *|<-l*cos|------| ||
| \ l / | || || | | || \ l / ||
| | || || 3 | | ||--------------- otherwise||
/ | || || x | | \\ pi*n /|
| || || -- otherwise | | |
| || \\ 3 / | |
| ||----------------------------------------------------------------------------- otherwise| |
\ \\ pi*n / /
$$\int l^{2} x^{3} t \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + l^{2} t \left(x^{3} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \cos{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \left(\begin{cases} - \frac{2 l^{3} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi^{3} n^{3}} + \frac{2 l^{2} x \cos{\left(\frac{\pi n x}{l} \right)}}{\pi^{2} n^{2}} + \frac{l x^{2} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{for}\: n \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)\right)$$
Respuesta
[src]
/ / /pi*n\ 4 /pi*n\ 2 /pi*n\ 3 /pi*n\\ | | l*cos|----| 6*l *sin|----| 3*l *sin|----| 6*l *cos|----|| | 2 | \ l / \ l / \ l / \ l /| |t*l *|- ----------- - -------------- + -------------- + --------------| for And(n > -oo, n < oo, n != 0) < | pi*n 4 4 2 2 3 3 | | \ pi *n pi *n pi *n / | | 0 otherwise \
$$\begin{cases} l^{2} t \left(- \frac{6 l^{4} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{4} n^{4}} + \frac{6 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{3 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right) & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ / /pi*n\ 4 /pi*n\ 2 /pi*n\ 3 /pi*n\\ | | l*cos|----| 6*l *sin|----| 3*l *sin|----| 6*l *cos|----|| | 2 | \ l / \ l / \ l / \ l /| |t*l *|- ----------- - -------------- + -------------- + --------------| for And(n > -oo, n < oo, n != 0) < | pi*n 4 4 2 2 3 3 | | \ pi *n pi *n pi *n / | | 0 otherwise \
$$\begin{cases} l^{2} t \left(- \frac{6 l^{4} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{4} n^{4}} + \frac{6 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{3 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right) & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((t*l^2*(-l*cos(pi*n/l)/(pi*n) - 6*l^4*sin(pi*n/l)/(pi^4*n^4) + 3*l^2*sin(pi*n/l)/(pi^2*n^2) + 6*l^3*cos(pi*n/l)/(pi^3*n^3)), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.